12 problems

Scaling-limit conjecture. As $n\to\infty$, the following limits hold. If $0\leq\alpha_+<\tilde\alpha_+$ and $0\leq\alpha_-<\tilde\alpha_-$ are fixed, then

Mean identity conjecture. For fixed parameters $p_+$ and $p_-$,

Let $L(\sigma)$ denote the statistic defined for the wreath product action, with $\sigma$ sampled from the relevant wreath product of parameter $k=2$. Wreath product longest increa…

Let $(\tau_n)_n$ be a sequence of conjugation-invariant random permutations, where $\tau_n$ has size $n$ and random cycle type $t^{(n)}$. Suppose that … in probability. Conjecture…

Let $L_n^{\varoast}$ denote the relevant longest monotone subsequence length for $\varoast\in\{\boxslash,\boxbslash,\boxdot\}$, let $\gamma(\varoast)$ and $\delta(\varoast)$ be the…

Let $t_0<t_1$ be any ordered pair of real numbers. For the probability $p_{\boxdot}(n;l)$ and the scaling variable $t_{l+1}(n)$, suppose $F_1$ and the functions $F_{\boxdot,j}$ are…

Let $L_n$ denote the length of the longest increasing subsequence in a random permutation of $n$ elements, and let $F_2$ be the Tracy–Widom distribution for the Gaussian unitary en…

Let $l_N^{\boxbslash}$ and $l_N^{\boxslash}$ denote the lengths of the two symmetric longest increasing subsequences considered in the source, and let…

Let $l_N^\square$ be the length of the longest increasing subsequence in the finite $N$ model, and let $E_2^{\rm soft}(0;(t,\infty))$ denote the soft-edge limiting distribution. Se…

Let $L_n$ denote the length of the longest increasing subsequence of a random walk with step lengths drawn from a distribution of finite variance. The expectation $\mathbb{E}(L_n)$…

Let $L_n$ denote the length of the longest increasing subsequence of an $n$-step random walk whose step lengths have finite variance. Finite-variance scaling conjecture. The correc…

Fluctuation conjecture. If $0<\alpha<\frac12$, then